A212891 Rectangular array: (row n) = b**c, where b(h) = h, c(h) = (n-1+h)^2, n>=1, h>=1, and ** = convolution.

1, 6, 4, 20, 17, 9, 50, 46, 34, 16, 105, 100, 84, 57, 25, 196, 190, 170, 134, 86, 36, 336, 329, 305, 260, 196, 121, 49, 540, 532, 504, 450, 370, 270, 162, 64, 825, 816, 784, 721, 625, 500, 356, 209, 81, 1210, 1200, 1164, 1092, 980, 830, 650, 454, 262

Offset: 1

Author

Clark Kimberling, Jun 16 2012

Keywords

• nonn
• easy
• tabl

Comments

Principal diagonal: A213436

Antidiagonal sums: A024166

row 1, (1,2,3,...)**(1,4,9,...): A002415(k+1)

row 2, (1,2,3,...)**(4,9,16,...): k*(k^3 + 8*k^2 + 23*k + 16)/12

row 3, (1,2,3,...)**(9,16,25,...): k*(k^3 + 12*k^2 + 53*k + 42)/12

...

For a guide to related arrays, see A213500.

Examples

			Northwest corner (the array is read by falling antidiagonals):
1....6....20....50....105....196...336
4....17...46....100...190....329...532
9....34...84....170...305....504...784
16...57...134...260...450....721...1092
25...86...196...370...625....980...1456
...
T(5,1) = (1)**(25) = 25
T(5,2) = (1,2)**(25,36) = 1*36+2*25 = 86
T(5,3) = (1,2,3)**(25,36,49) = 1*49+2*36+3*25 = 196
		

Crossrefs

Cf. A213500.

Programs

Mathematica

b[n_] := n; c[n_] := n^2
t[n_, k_] := Sum[b[k - i] c[n + i], {i, 0, k - 1}]
TableForm[Table[t[n, k], {n, 1, 10}, {k, 1, 10}]]
Flatten[Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}]]
r[n_] := Table[t[n, k], {k, 1, 60}]  (* A212891 *)
d = Table[t[n, n], {n, 1, 40}] (* A213436 *)
s[n_] := Sum[t[i, n + 1 - i], {i, 1, n}]
s1 = Table[s[n], {n, 1, 50}] (* A024166  *)

Formula

T(n,k) = 5*T(n,k-1) - 10*T(n,k-2) + 10*T(n,k-3) - 5*T(n,k-4) + T(n,k-5).

G.f. for row n: f(x)/g(x), where f(x) = n^2 - (2*n^2 - 2*n - 1)*x + ((n-1)^2)*x^2 and g(x) = (1 - x)^5.